Optimal. Leaf size=54 \[ -\frac {x}{2}-\frac {1}{2} \sqrt {2-x} \sqrt {x}-\frac {1}{2} \log (1-x)+\frac {1}{2} \tanh ^{-1}\left (\sqrt {2-x} \sqrt {x}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2105, 101, 12, 92, 206, 43} \begin {gather*} -\frac {x}{2}-\frac {1}{2} \sqrt {2-x} \sqrt {x}-\frac {1}{2} \log (1-x)+\frac {1}{2} \tanh ^{-1}\left (\sqrt {2-x} \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 92
Rule 101
Rule 206
Rule 2105
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{\sqrt {2-x}-\sqrt {x}} \, dx &=\int \frac {\sqrt {2-x} \sqrt {x}}{2-2 x} \, dx+\int \frac {x}{2-2 x} \, dx\\ &=-\frac {1}{2} \sqrt {2-x} \sqrt {x}+\frac {1}{2} \int \frac {2}{(2-2 x) \sqrt {2-x} \sqrt {x}} \, dx+\int \left (-\frac {1}{2}-\frac {1}{2 (-1+x)}\right ) \, dx\\ &=-\frac {1}{2} \sqrt {2-x} \sqrt {x}-\frac {x}{2}-\frac {1}{2} \log (1-x)+\int \frac {1}{(2-2 x) \sqrt {2-x} \sqrt {x}} \, dx\\ &=-\frac {1}{2} \sqrt {2-x} \sqrt {x}-\frac {x}{2}-\frac {1}{2} \log (1-x)+2 \operatorname {Subst}\left (\int \frac {1}{4-4 x^2} \, dx,x,\sqrt {2-x} \sqrt {x}\right )\\ &=-\frac {1}{2} \sqrt {2-x} \sqrt {x}-\frac {x}{2}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {2-x} \sqrt {x}\right )-\frac {1}{2} \log (1-x)\\ \end {align*}
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Mathematica [A] time = 0.14, size = 82, normalized size = 1.52 \begin {gather*} \frac {1}{2} \left (-x-\sqrt {-((x-2) x)}-\log \left (1-\sqrt {x}\right )-\log \left (\sqrt {x}+1\right )+\tanh ^{-1}\left (\frac {2-\sqrt {x}}{\sqrt {2-x}}\right )-\tanh ^{-1}\left (\frac {\sqrt {x}+2}{\sqrt {2-x}}\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [C] time = 0.12, size = 76, normalized size = 1.41 \begin {gather*} -\frac {x}{2}-\frac {1}{2} \sqrt {2-x} \sqrt {x}-\log \left (\sqrt {2-x}-i \sqrt {x}+(1-i)\right )-2 \tanh ^{-1}\left ((-1-i) \sqrt {2-x}-(1-i) \sqrt {x}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 64, normalized size = 1.19 \begin {gather*} -\frac {1}{2} \, x - \frac {1}{2} \, \sqrt {x} \sqrt {-x + 2} - \frac {1}{2} \, \log \left (x - 1\right ) + \frac {1}{2} \, \log \left (\frac {x + \sqrt {x} \sqrt {-x + 2}}{x}\right ) - \frac {1}{2} \, \log \left (-\frac {x - \sqrt {x} \sqrt {-x + 2}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 51, normalized size = 0.94 \begin {gather*} -\frac {x}{2}-\frac {\ln \left (x -1\right )}{2}-\frac {\sqrt {-x +2}\, \left (-\arctanh \left (\frac {1}{\sqrt {-\left (x -2\right ) x}}\right )+\sqrt {-\left (x -2\right ) x}\right ) \sqrt {x}}{2 \sqrt {-\left (x -2\right ) x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {\sqrt {x}}{\sqrt {x} - \sqrt {-x + 2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 56, normalized size = 1.04 \begin {gather*} \mathrm {atanh}\left (\frac {\sqrt {x}\,\left (\sqrt {2}-\sqrt {2-x}\right )}{x+\sqrt {2}\,\sqrt {2-x}-2}\right )-\frac {\ln \left (x-1\right )}{2}-\frac {x}{2}-\frac {\sqrt {x}\,\sqrt {2-x}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x}}{- \sqrt {x} + \sqrt {2 - x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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